# Evaluation AG code

## Description

Also called a function code. Evaluation code over \(GF(q)\) on a set of points \({\cal P} = \left( P_1,P_2,\cdots,P_n \right)\) in \(GF(q)\) lying on an algebraic curve \(\cal X\) whose corresponding vector space \(L\) of functions \(f\) consists of certain polynomials or rational functions. Codewords are evaluations of all functions at the specified points, \begin{align} \left( f(P_1), f(P_2), \cdots, f(P_n) \right) \quad\quad\forall f\in L~. \tag*{(1)}\end{align} The code is denoted as \(C_L({\cal X},{\cal P},D)\), where the divisor \(D\) (of degree less than \(n\)) determines which rational functions to use by prescribing features associated with their zeroes and poles. The original motivation for evaluation codes, which are generalizations of RS codes that expand both the types of functions used as well as the available evaluation points, was to increase code length while maintaining good distance and size.

The algebraic curve \(\cal X\) used for this construction is the set of zeroes of a nontrivial polynomial that is both smooth and irreducible over any field extension of \(GF(q)\). The curve can be defined over affine space or projective space, which contains the affine coordinates as a subset and which can yield an increase in length. If evaluations are made over projective coordinates, then the codewords are evaluations of homogeneous polynomials, and there are relations between such polynomials with polynomials over affine space. See Refs. [1][2] for more details.

In the case of polynomial functions \(f\), evaluation AG codes reduce to polynomial evaluation codes on algebraic curves. In the general case of rational functions, which are ratios of two polynomials, one can select such features for both the numerator and denominator polynomials. Zeroes of the denominator polynomial are called poles of the rational function, and their multiplicities correspond to orders of the poles. A bookkeeping device for this data is the divisor \(D\), and the corresponding vector space of functions defined using the curve \(\cal X\) and the divisor is the Riemann-Roch space \(L=L(D)\) ([3], pg. 313).

## Protection

## Decoding

## Notes

## Parents

- Evaluation code — Evaluation AG codes are evaluation codes of rational functions \(f\) for which \(\cal X\) is an algebraic curve, i.e., an algebraic variety of dimension one [1].
- Algebraic-geometry (AG) code

## Children

- Elliptic code — Elliptic codes are evaluation AG codes with \(\cal X\) being an elliptic curve, i.e., curve of genus one ([23], Ch. 3.2; [1]).
- Hermitian code — Hermitian codes are evaluation AG codes with \(\cal X\) being a Hermitian curve ([1], Ex. 2.74). This curve is maximal, meaning that Hermitian codes are evaluation AG codes with maximum possible length given a fixed genus.
- Klein-quartic code — Klein-quartic codes are evaluation AG codes with \(\cal X\) being the Klein quartic ([1], Ex. 2.75).
- Plane-curve code — Plane-curve codes are evaluation AG codes of bivariate polynomials with \(\cal X\) being an affine plane curve ([1], Thm. 2.27).
- Suzuki-curve code — Suzuki-curve codes are evaluation AG codes with \(\cal X\) being a Suzuki curve.
- Residue AG code — Any residue AG code of differential forms can be equivalently stated as an evaluation AG code of functions ([3], Lemma 15.3.10; [1], Thm. 2.72). In addition, evaluation and residue AG codes are dual to each other ([3], pg. 313; [1]).
- Generalized RS (GRS) code — GRS (RS) codes are in one-to-one correspondence with evaluation AG codes of univariate polynomials \(f\) with \(\cal X\) being the projective (affine) line ([3], Thm. 15.3.24; [23], Ch. 3.2; [1]).
- Hexacode — The hexacode is an evaluation AG code over \(GF(4) = \{0,1,\omega, \bar{\omega}\}\) with \(\cal X\) defined by \(x^2 y + \omega y^2 z + \bar{\omega} z^2 x = 0\) ([1], Ex. 2.77).

## Cousins

- Linear \(q\)-ary code — The degree of the divisor for evaluation AG codes is restricted to be less than \(n\). When there is no restriction, any \(q\)-ary linear code can be formulated as an evaluation AG code [24].
- Polynomial evaluation code — Evaluation AG codes are evaluation codes on algebraic curves. Polynomial evaluation codes are evaluation codes of polynomials. Evaluation AG codes of polynomials are equivalent to polynomial evaluation codes on algebraic curves.

## References

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## Page edit log

- Victor V. Albert (2022-08-11) — most recent
- Victor V. Albert (2022-03-22)

## Cite as:

“Evaluation AG code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/evaluation